A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations

In this paper,we construct a high-order discontinuous Galerkin(DG)method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics(VRMHD).To control the divergence error in the magnetic field,both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD.Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes,respectively,showing that the scheme can maintain the positivity-preserving(PP)property under some CFL conditions when combined with the strong-stability-preserving time discretization.Then,general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes.Numerical tests demonstrate the effectiveness of the proposed schemes.

High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws

In this paper,the maximum-principle-preserving(MPP)and positivitypreserving(PP)flux limiting technique will be generalized to a class of high-order weighted compact nonlinear schemes(WCNSs)for scalar conservation laws and the compressible Euler systems in both one and two dimensions.The main idea of the present method is to rewrite the scheme in a conservative form,and then define the local limiting parameters via case-by-case discussion.Smooth test problems are presented to demonstrate that the proposed MPP/PP WCNSs incorporating a third-order Runge-Kutta method can attain the desired order of accuracy.Other test problems with strong shocks and high pressure and density ratios are also conducted to testify the performance of the schemes.

A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes

Two-dimensional three-temperature(2-D 3-T)radiation diffusion equa-tions are widely used to approximately describe the evolution of radiation energy within a multimaterial system and explain the exchange of energy among electrons,ions and photons.In this paper,we suggest a new positivity-preserving finite volume scheme for 2-D 3-T radiation diffusion equations on general polygonal meshes.The vertex unknowns are treated as primary ones for which the finite volume equations are constructed.The edgemidpoint and cell-centered unknowns are used as auxiliary ones and interpolated by the primary unknowns,which makes the final scheme a pure vertex-centered one.By comparison,most existing positivity-preserving finite volume schemes are cell-centered and based on the convex decomposition of the co-normal.Here,the conormal decomposition is not convex in general,leading to a fixed stencil of the flux approximation and avoiding a certain search algo-rithm on complex grids.Moreover,the new scheme effectively alleviates the nu-merical heat-barrier issue suffered by most existing cell-centered or hybrid schemes in solving strongly nonlinear radiation diffusion equations.Numerical experiments demonstrate the second-order accuracy and the positivity of the solution on various distorted grids.For the problem without analytic solution,the contours of the nu-merical solutions obtained by our scheme on distorted meshes accord with those on smooth quadrilateral meshes.

A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations

A high-order, well-balanced, positivity-preserving quasi-Lagrange movingmesh DG method is presented for the shallow water equations with non-flat bottomtopography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake ortsunami waves in the deep ocean. The method combines a quasi-Lagrange movingmesh DG method, a hydrostatic reconstruction technique, and a change of unknownvariables. The strategies in the use of slope limiting, positivity-preservation limiting,and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treatsmesh movement continuously in time and has the advantages that it does not need tointerpolate flow variables from the old mesh to the new one and places no constraintfor the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the wellbalance property, positivity preservation, and high-order accuracy of the method andits ability to adapt the mesh according to features in the flow and bottom topography.

A Positivity-Preserving Scheme for the Simulation of Streamer Discharges in Non-Attaching and Attaching Gases

Assumed having axial symmetry,the streamer discharge is often described by a fluid model in cylindrical coordinate system,which consists of convection dominated(diffusion)equations with source terms,coupled with a Poisson’s equation.Without additional care for a stricter CFL condition or special treatment to the negative source term,popular methods used in streamer discharge simulations,e.g.,FEMFCT,FVM,cannot ensure the positivity of the particle densities for the cases in attaching gases.By introducing the positivity-preserving limiter proposed by Zhang and Shu[15]and Strang operator splitting,this paper proposes a finite difference scheme with a provable positivity-preserving property in cylindrical coordinate system,for the numerical simulation of streamer discharges in non-attaching and attaching gases.Numerical examples in non-attaching gas(N_(2))and attaching gas(SF_(6))are given to illustrate the effectiveness of the scheme.

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

In order to suppress the failure of preserving positivity of density or pres-sure,a positivity-preserving limiter technique coupled with h-adaptive Runge-Kutta discontinuous Galerkin(RKDG)method is developed in this paper.Such a method is implemented to simulate flows with the large Mach number,strong shock/obstacle interactions and shock diffractions.The Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is also presented.This ap-proach directly uses the cell solution polynomial of DG finite element space as the interpolation formula.The method is validated by the well documented test ex-amples involving unsteady compressible flows through complex bodies over a large Mach numbers.The numerical results demonstrate the robustness and the versatility of the proposed approach.

Third-order unconditional positivity-preserving schemes for reactive flows keeping both mass and mole balance

In this paper,the previously proposed second-order process-based modified Patankar Runge-Kutta schemes are extended to the third order of accuracy.Owing to the process-based implicit handling of reactive source terms,the mass conservation,mole balance and energy conservation are kept simultaneously while the positivity for the density and pressure is preserved unconditionally even with stiff reaction networks.It is proved that the first-order truncation terms for the Patankar coefficients must be zero to achieve a prior third order of accuracy for most cases.A twostage Patankar procedure for each Runge-Kutta step is designed to eliminate the first-order truncation terms,accomplish the prior third order of accuracy and maximize the Courant number which the total variational diminishing property requires.With the same approach as the second-order schemes,the third-order ones are applied to Euler equations with chemical reactive source terms.Numerical studies including both 1D and 2D ordinary and partial differential equations are conducted to affirm both the prior order of accuracy and the positivity-preserving property for the density and pressure.

High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes

In this paper,we propose a high-order accurate discontinuous Galerkin(DG)method for the compressible Euler equations under gravitationalfields on un-structured meshes.The scheme preserves a general hydrostatic equilibrium state and provably guarantees the positivity of density and pressure at the same time.Compar-ing with the work on the well-balanced scheme for Euler equations with gravitation on rectangular meshes,the extension to triangular meshes is conceptually plausible but highly nontrivial.Wefirst introduce a special way to recover the equilibrium state and then design a group of novel variables at the interface of two adjacent cells,which plays an important role in the well-balanced and positivity-preserving properties.One main challenge is that the well-balanced schemes may not have the weak positivity property.In order to achieve the well-balanced and positivity-preserving properties simultaneously while maintaining high-order accuracy,we carefully design DG spa-tial discretization with well-balanced numericalfluxes and suitable source term ap-proximation.For the ideal gas,we prove that the resulting well-balanced scheme,cou-pled with strong stability preserving time discretizations,satisfies a weak positivity property.A simple existing limiter can be applied to enforce the positivity-preserving property,without losing high-order accuracy and conservation.Extensive one-and two-dimensional numerical examples demonstrate the desired properties of the pro-posed scheme,as well as its high resolution and robustness.

A POSITIVITY-PRESERVING FINITE ELEMENT METHOD FOR QUANTUM DRIFT-DIFFUSION MODEL

In this paper,we propose a positivity-preserving finite element method for solving the three-dimensional quantum drift-diffusion model.The model consists of five nonlinear elliptic equations,and two of them describe quantum corrections for quasi-Fermi levels.We propose an interpolated-exponential finite element(IEFE)method for solving the two quantum-correction equations.The IEFE method always yields positive carrier densities and preserves the positivity of second-order differential operators in the Newton linearization of quantum-correction equations.Moreover,we solve the two continuity equations with the edge-averaged finite element(EAFE)method to reduce numerical oscillations of quasi-Fermi levels.The Poisson equation of electrical potential is solved with standard Lagrangian finite elements.We prove the existence of solution to the nonlinear discrete problem by using a fixed-point iteration and solving the minimum problem of a new discrete functional.A Newton method is proposed to solve the nonlinear discrete problem.Numerical experiments for a three-dimensional nano-scale FinFET device show that the Newton method is robust for source-to-gate bias voltages up to 9V and source-to-drain bias voltages up to 10V.

A High Order Positivity-Preserving Discontinuous Galerkin Remapping Method Based on a Moving Mesh Solver for ALE Simulation of the Compressible Fluid Flow

The arbitrary Lagrangian-Eulerian(ALE)method is widely used in the field of compressible multi-material and multi-phase flow problems.In order to implement the indirect ALE approach for the simulation of compressible flow in the context of high order discontinuous Galerkin(DG)discretizations,we present a high order positivity-preserving DG remapping method based on a moving mesh solver in this paper.This remapping method is based on the ALE-DG method developed by Klingenberg et al.[17,18]to solve the trivial equation∂u/∂t=0 on a moving mesh,which is the old mesh before remapping at t=0 and is the new mesh after remapping at t=T.An appropriate selection of the final pseudo-time T can always satisfy the relatively mild smoothness requirement(Lipschitz continuity)on the mesh movement velocity,which guarantees the high order accuracy of the remapping procedure.We use a multi-resolution weighted essentially non-oscillatory(WENO)limiter which can keep the essentially non-oscillatory property near strong discontinuities while maintaining high order accuracy in smooth regions.We further employ an effective linear scaling limiter to preserve the positivity of the relevant physical variables without sacrificing conservation and the original high order accuracy.Numerical experiments are provided to illustrate the high order accuracy,essentially non-oscillatory performance and positivity-preserving of our remapping algorithm.In addition,the performance of the ALE simulation based on the DG framework with our remapping algorithm is examined in one-and two-dimensional Euler equations.

Stable Runge-Kutta discontinuous Galerkin solver for hypersonic rarefied gaseous flow based on 2D Boltzmann kinetic model equations

A stable high-order Runge-Kutta discontinuous Galerkin(RKDG) scheme that strictly preserves positivity of the solution is designed to solve the Boltzmann kinetic equation with model collision integrals. Stability is kept by accuracy of velocity discretization, conservative calculation of the discrete collision relaxation term, and a limiter. By keeping the time step smaller than the local mean collision time and forcing positivity values of velocity distribution functions on certain points, the limiter can preserve positivity of solutions to the cell average velocity distribution functions. Verification is performed with a normal shock wave at a Mach number 2.05, a hypersonic flow about a two-dimensional(2D) cylinder at Mach numbers 6.0 and 12.0, and an unsteady shock tube flow. The results show that, the scheme is stable and accurate to capture shock structures in steady and unsteady hypersonic rarefied gaseous flows. Compared with two widely used limiters, the current limiter has the advantage of easy implementation and ability of minimizing the influence of accuracy of the original RKDG method.

High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes

A high order finite difference numerical scheme is developed for the shallow water equations on curvilinear meshes based on an alternative flux formulation of the weighted essentially non-oscillatory(WENO)scheme.The exact C-property is investigated,and comparison with the standard finite difference WENO scheme is made.Theoretical derivation and numerical results show that the proposed finite difference WENO scheme can maintain the exact C-property on both stationarily and dynamically generalized coordinate systems.The Harten-Lax-van Leer type flux is developed on general curvilinear meshes in two dimensions and verified on a number of benchmark problems,indicating smaller errors compared with the Lax-Friedrichs solver.In addition,we propose a positivity-preserving limiter on stationary meshes such that the scheme can preserve the non-negativity of the water height without loss of mass conservation.

Simple and robust h-adaptive shock-capturing method for flux reconstruction framework

In this paper,a simple and robust shock-capturing method is developed for the Flux Reconstruction(FR)framework by combining the Adaptive Mesh Refinement(AMR)technique with the positivity-preserving property.The adaptive technique avoids the use of redundant meshes in smooth regions,while the positivity-preserving property makes the solver capable of providing numerical solutions with physical meaning.The compatibility of these two significant features relies on a novel limiter designed for mesh refinements.It ensures the positivity of solutions on all newly created cells.Therefore,the proposed method is completely positivity-preserving and thus highly robust.It performs well in solving challenging problems on highly refined meshes and allows the transition of cells at different levels to be completed within a very short distance.The performance of the proposed method is examined in various numerical experiments.When solving Euler equations,the technique of Local Artificial Diffusivity(LAD)is additionally coupled to damp oscillations.More importantly,when solving Navier-Stokes equations,the proposed method requires no auxiliaries and can provide satisfying numerical solutions directly.The implementation of the method becomes rather simple.

Monotonicity Correction for the Finite Element Method of Anisotropic Diffusion Problems

We apply the monotonicity correction to thefinite element method for the anisotropic diffusion problems,including linear and quadraticfinite elements on triangular meshes.When formulating thefinite element schemes,we need to calculate the integrals on every triangular element,whose results are the linear combination of the two-point pairs.Then we decompose the integral results into the main and remaining parts according to coefficient signs of two-point pairs.We apply the nonlinear correction to the positive remaining parts and move the negative remaining parts to the right side of thefinite element equations.Finally,the original stiffness matrix can be transformed into a nonlinear M-matrix,and the corrected schemes have the positivity-preserving property.We also give the monotonicity correction to the time derivative term for the time-dependent problems.Numerical experiments show that the correctedfinite element method has monotonicity and maintains the convergence order of the original schemes in H1-norm and L2-norm,respectively.

A Hybrid Numerical Simulation of Supersonic IsotropicTurbulence

This paper presents an extension work of the hybrid scheme proposed by Wang et al.[J.Comput.Phys.229(2010)169-180]for numerical simulation of sub-sonic isotropic turbulence to supersonic turbulence regime.The scheme still utilizes an 8th-order compact scheme with built-in hyperviscosity for smooth regions and a 7th-order WENO scheme for highly compression regions,but now both in their con-servation formulations and for the latter with the Roe type characteristic-wise recon-struction.To enhance the robustness of the WENO scheme without compromising its high-resolution and accuracy,the recursive-order-reduction procedure is adopted,where a new type of reconstruction-failure-detection criterion is constructed from the idea of positivity-preserving.In addition,a new form of cooling function is proposed,which is proved also to be positivity-preserving.With a combination of these techniques,the new scheme not only inherits the good properties of the original one but also extends largely the computable range of turbulent Mach number,which has been further confirmed by numerical results.

Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates

In this paper,we analyze the explicit Runge-Kutta discontinuous Galerkin(RKDG)methods for the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology.The RKDG methods use a third order explicit total-variation-diminishing Runge-Kutta(TVDRK3)time discretization and upwinding numerical fluxes.By using the energy method,under a standard CourantFriedrichs-Lewy(CFL)condition,we obtain L2stability for general solutions and a priori error estimates when the solutions are smooth enough.The theoretical results are proved for piecewise polynomials with any degree k 1.Finally,since the solutions to this system are non-negative,we discuss a positivity-preserving limiter to preserve positivity without compromising accuracy.Numerical results are provided to demonstrate these RKDG methods.

非结构网格间断Galerkin方法的一种简单有效后验激波指示器

针对间断Galerkin(DG)方法的激波捕捉问题,在三角形网格上提出了一种基于后验方法的简单有效的问题单元指示器.方法利用了von Neumann单元中不同时刻的离散解来保持DG格式的紧凑性.本文采用有效的技术来提供有关问题单元的进一步信息,并且可以对不同的问题单元应用不同的限制方法,因此可以获得良好的数值特性,包括保正性质和振荡抑制性质.本文采用了TVB限制器和Hermite WENO限制器,对比了提出的指示器与TVB指示器的性能.数值结果表明了当前后验激波指示器的有效性和鲁棒性.